Joint distributions
01
Finish a PMF table - Strange families
Suppose that 15 percent of the families in a strange community have no children, 20 percent have 1 child, 35 percent have 2 children, and 30 percent have 3 children. Assume the odds of a child being a boy or a girl are equal.
If a family is chosen at random from this community, then
, the number of boys, and , the number of girls, in this family will have the joint PMF partially shown in Table 6.2:
(a) Complete the table by finding the missing entries.
(b) What is the probability that “
or is 1”?
02
PMF calculations from a table
Suppose the joint PMF of
and has values given in this table:
0 1 2 3 1 0.10 0.15 0 0.05 2 0.20 0.05 0.05 0.20 3 0.05 0 0.05
- (a) Find
. - (b) Find the marginal PMF of
. - (c) Find the PMF of the random variable
. - (d) Find
and .
03
Marginals from joint PMF
Suppose the discrete joint PMF of
and is given by:
Compute the marginal PMFs
and .
04
Joint CDF on box events: All four corners
Consider the following formula:
Prove this formula. Hint: Do these steps along the way:
Draw these events in the
-plane: Draw the event
. Write the probability of this event in terms of .
05
Marginals from PDF
Suppose
and have joint PDF given by:
- (a) Find the marginal PDFs for
and . - (b) Find
.
Independent random variables
06
Random point in a triangle
Consider a joint distribution whose PDF is constant inside the triangle with vertices
, and , and zero outside. Suppose a point is chosen at random according to this distribution.
- (a) Find the joint PDF
. - (b) Find the marginal PDFs for
and . - (c) Are
and independent?
07
Factorizing the density Consider two joint density functions for
and : (Assume the densities are zero outside the given domain.)
Supposing
is the joint density, are and independent? Why or why not? Supposing is the joint density, are and independent? Why or why not?
08
Composite PDF from joint PDF The joint density of random variables
and is given by: Compute the PDF of
. (Hint: First find the CDF of .)